510 
Through the points P of space the polar planes a, , a, with 
respect to ®,, db, are conjugated one by one to each other; so we 
can regard the lines s as the lines of intersection of conjugated planes 
of two collinear spaces, and we then find immediately that the lines 
s form’) a tetrahedral complee, for which the tetrahedron of the 
four vertices of the cones of k* is the surface of singularity, in such 
a sense that each arbitrary ray through one of the vertices or in one 
of the faces of that tetrahedron is a complex ray, whilst in general 
the tetrahedral complex being quadratic a point has but a quadratic 
complex cone, a plane a quadratic complex curve. As namely the two 
polar planes of the vertex of a cone coincide in the opposite face of the 
tetrahedron, each line in this face can be regarded asa ray s, and as 
of a line 7 through 7, e. g. the t wo conjugated lines lie in 7,7, 7, 
inversely the two polar planes of the point of intersection of those 
conjugated lines pass through /, so that / is a complex ray s. The 
complex cone of a point P in 7,7,7,==rt, breaks up into two 
pencils, one with vertex P and lying in 1,, the other with vertex Ye 
and lying in a certain plane through P and 7’,; and likewise the 
complex curve in a plane through 7, degenerates into 2 points, viz. 
T, itself and a certain point in the line of intersection of that plane and r. 
A ray s being the line of intersection of the polar planes 2, , x, of 
a certain point / with respect to ®, , ®,, inversely through an arbitrary 
ray s two planes x, , zr, must pass having the same pole ?; if however 
a line lies in a plane, then the conjugated line passes through the pole 
of that plane; thus for s the fro conjugated lines s,,s, must pass 
through P and must intersect each other in P; so we can also define 
the rays s as those rays of space whose two conjugated lines with 
respect to ®,, ®, intersect each other. In this we have also a means 
to determine the focus of an arbitrary ray s; we have but to find 
the point of intersection of s, and s,. 
The rays s conjugated to the points of an arbitrary line / form 
a regulus as we have seen above; those conjugated to the points of 
a ray s must thus form according to the preceding a quadratic cone, 
and this is evidently the complex cone for the focus P of s, by 
means of which a construction for that cone has been found; we 
take the ray s conjugated to ?, we allow a point to describe that 
ray and we determine for each position ihe two polar planes; the 
line of intersection of these describes the complex cone when the point 
describes the ray s. Just as the regulus for a line /, so each com- 
plex cone contains the vertices of the four doubly projecting cones ; 
1) Sturm |. c. p. 342, 
